J Dyn Diff Equat (2014) 26:583–605DOI 10.1007/s10884-014-9355-4

Traveling Wave Solutions for Delayed Reaction–DiffusionSystems and Applications to Diffusive Lotka–VolterraCompetition Models with Distributed Delays

Guo Lin · Shigui Ruan

Received: 24 October 2012 / Revised: 29 January 2014 / Published online: 6 March 2014© Springer Science+Business Media New York 2014

Abstract This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wavesolutions is reduced to the existence of generalized upper and lower solutions. Using thetechnique of contracting rectangles, the asymptotic behavior of traveling wave solutions fordelayed diffusive systems is obtained. To illustrate our main results, the existence, nonexis-tence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterracompetition systems with distributed delays are established. The existence of nonmonotonetraveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed.In particular, it is proved that if there exists instantaneous self-limitation effect, then the largedelays appearing in the intra-specific competitive terms may not affect the existence andasymptotic behavior of traveling wave solutions.

Keywords Nonmonotone traveling wave solutions · Contracting rectangle · Invariantregion · Generalized upper and lower solutions

Mathematics Subject Classification 35C07 · 37C65 · 35K57

1 Introduction

In studying the nonlinear dynamics of delayed reaction–diffusion systems, one of the impor-tant topics is the existence of traveling wave solutions because of their significant roles inbiological invasion and epidemic spreading, we refer to Ai [2], Gourley and Ruan [4], Fangand Wu [5], Faria et al. [6], Faria and Trofimchuk [7,8], Huang and Zou [12], Kwong and Ou

G. Lin (B)School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000,Gansu, People’s Republic of Chinae-mail: ling@lzu.edu.cn

S. RuanDepartment of Mathematics, University of Miami, P. O. Box 249085, Coral Gables, FL 33124-4250, USAe-mail: ruan@math.miami.edu

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[14], Li et al. [15], Liang and Zhao [16], Lin et al. [17], Ma [18,19], Ma and Wu [20], Mei[22], Ou and Wu [23], Schaaf [26], Smith and Zhao [29], Thieme and Zhao [31], Wang [33],Wang et al. [34–36], Wu and Zou [38], and Yi et al. [40]. It has been shown that delay mayinduce some differences of traveling wave solutions between the delayed and undelayed sys-tems, for example, the minimal wave speed (see Schaaf [26], Zou [41]) and the monotonicityof traveling wave solutions in scalar equations (see an example in Faria and Trofimchuk [7]).In particular, the asymptotic behavior of traveling wave solutions, which is often formulatedby the asymptotic boundary conditions, plays a very crucial role since it describes the prop-agation processes in different natural environments. For example, with proper asymptoticboundary value conditions, the traveling wave solutions of two species diffusive competitionsystems may reflect the coinvasion-coexistence of two invaders (see Ahmad and Lazer [1],Li et al. [15], Lin et al. [17], Tang and Fife [30]), or exclusion between an invader and aresident (see Gourley and Ruan [4], Huang [13]). Therefore, understanding the asymptoticbehavior is a fundamental issue in the study of traveling wave solutions. Moreover, verydetailed asymptotic behavior of traveling wave solutions has also been studied due to thedevelopment in mathematical theory of delayed reaction–diffusion systems, such as in theasymptotic stability and uniqueness of traveling wave solutions (see Mei et al. [22], Volpertet al. [32], Wang et al. [36]).

To obtain the asymptotic behavior of traveling wave solutions in delayed reaction–diffusion systems, there are several methods. The first is based on the monotonicity of trav-eling wave solutions, which is often considered under the assumption of quasimonotonicityin the sense of proper ordering (Huang and Zou [12], Ma [18], Wang et al. [34], Wu andZou [38]). The second is to construct proper auxiliary functions, such as the upper and lowersolutions (Li et al. [15], Lin et al. [17]). When these methods fail, a fluctuation technique isutilized to study the asymptotic behavior of traveling wave solutions if the system satisfies thelocally quasimonotone condition near the unstable steady state (Ma [19], Wang [33]). Someother results have also been presented for (fast) traveling wave solutions of scalar equations(Faria and Trofimchuk [7,8], Kwong and Ou [14]).

Of course, before considering the asymptotic behavior of traveling wave solutions ofdelayed systems, we must establish the existence of nontrivial traveling wave solutions. Toobtain the existence of traveling wave solutions of delayed systems, Wu and Zou [38] used amonotone iteration scheme if a delayed system is cooperative in the sense of proper ordering,see also Ma [18]. Li et al. [15] further considered the existence of traveling wave solutionsin competition systems by a cross iteration technique. Very recently, Ma [19] studied thetraveling wave solutions of a locally monotone delayed equation by constructing auxiliarymonotone equations, see also Wang [33], Yi et al. [40]. Moreover, by regarding the time delayas a parameter, the existence of traveling wave solutions was also studied by the perturbationmethod or Banach fixed point theorem, see Gourley and Ruan [4], Ou and Wu [23].

In this paper, we first study the existence and asymptotic behavior of traveling wavesolutions in the following delayed reaction–diffusion system

∂vi (x, t)

∂t= di�vi (x, t)+ fi (vt (x)), (1.1)

where x ∈ R, t > 0, v = (v1, v2, . . . , vn) ∈ Rn, d1, d2, . . . , dn are positive con-

stants, vt (x) := v(x, t + s), s ∈ [−τ, 0] and τ > 0 is the time delay, therefore,fi : C([−τ, 0],Rn) → R, here C([−τ, 0],Rn) is the space of continuous functions definedon [−τ, 0] and valued in R

n, which is a Banach space equipped with the supremum norm.To overcome the difficulty arising from the deficiency of comparison principle, we intro-

duce the definition of generalized upper and lower solutions of the corresponding wave system

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of delayed system (1.1). By using Schauder’s fixed point theorem, the existence of travelingwave solutions is reduced to the existence of generalized upper and lower solutions. Moti-vated by the idea of contracting rectangles in evolutionary systems, we establish an abstractconclusion on the asymptotic behavior of positive traveling wave solutions in general partialfunctional differential equations. Subsequently, we consider two scalar delayed equationswith diffusion in population dynamics (see Ma [19] and Zou [41]), which implies that ourmethods can also be applied to some well studied models.

In population dynamics, the following Lotka–Volterra reaction–diffusion system withdistributed delay has been widely studied

∂ui (x, t)

∂t= di�ui (x, t)+ ri ui (x, t)

⎡⎣1 −

n∑j=1

ci j

0∫

−τu j (x, t + s)dηi j (s)

⎤⎦ , (1.2)

in which i ∈ {1, 2, . . . , n} =: I, x ∈ � ⊆ Rk, k ∈ N, t > 0, u = (u1, u2, . . . , un) ∈

Rn, ui (x, t) denotes the density of the i-th competitor at time t and in location x ∈ �, di >

0, ri > 0, cii > 0 and ci j ≥ 0 are constants for i, j ∈ I, i �= j. We also suppose that

ηi j (s) is nondecreasing on [−τ, 0] and ηi j (0)− ηi j (−τ) = 1,

which will be imposed throughout the paper. Let

ai = ηi i (0)− ηi i (0−), i ∈ I,

and set

ηi i (s) ={ηi i (s), s ∈ [−τ, 0),

ηi i (0−), s = 0,ηi j (s) = ηi j (s), i, j ∈ I, i �= j.

Clearly, ai > 0 implies the existence of instantaneous self-limitation effect in populationdynamics.

The dynamics of (1.2) has been studied by several authors, for example, Gourley andRuan [4], Fang and Wu [5], Li et al. [15], Lin et al. [17], Martin and Smith [21] and Ruanand Wu [25]. More precisely, when � is a bounded domain and (1.2) is equipped with theNeumann boundary condition, Martin and Smith [21] proved that if the initial values of (1.2)are positive and

n∑j=1

ci j (c j j a j )−1 < 2, i ∈ I, (1.3)

then the unique mild solution to (1.2) satisfies

ui (x, t) → u∗i , t → ∞, i ∈ I, x ∈ �, (1.4)

hereafter u∗ = (u∗1, u∗

2, . . . , u∗n) is the unique spatially homogeneous positive steady state

of (1.2), of which the existence can be obtained by (1.3). It is well known that ai < 1 impliesthe existence of time delay in intra-specific competition, which often leads to some significantdifferences between the dynamics of delayed and undelayed models if the time delay is large.For example, the following Logistic and Hutchinson equations

du(t)

dt= u(t)(1 − u(t)),

du(t)

dt= u(t)(1 − u(t − τ)), τ > 0

exhibit dramatically different dynamics, we refer to Ruan [24] for detailed analysis on thesetwo equations and Hale and Verduyn Lunel [11] and Wu [37] for fundamental theories on

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delayed equations. However, (1.4) does not depend on the size of delays and the distributionof ηi i (s) for s ∈ [−τ, 0). Similar phenomena can be founded in the corresponding functionaldifferential equations, see Smith [28, Sect. 5.7].

In particular, Li et al. [15] and Lin et al. [17] have established the existence of travelingwave solutions to (1.2), which models the invasion-coexistence scenario of multiple com-petitors. However, these results only hold if there exists τ0 ∈ [0, τ ] small enough satisfying∫ 0−τ0

dηi i (s) = 1, i ∈ I, which ensures the so-called exponentially monotone conditionsuch that the upper and lower solutions are admissible. If τ0 is large, then we cannot applythese techniques and results to study the existence and asymptotic behavior of traveling wavesolutions of (1.2). In this paper, we shall consider the existence and further properties ofnontrivial traveling wave solutions of (1.2) with ai > 0, i ∈ I .

Using the results on the existence and asymptotic behavior of traveling wave solutionsin system (1.1), model (1.2) with � = R is studied by presenting the existence, nonex-istence and asymptotic behavior of positive traveling wave solutions. In particular, due toless requirements for auxiliary functions, we obtain some sufficient conditions on the exis-tence of nonmonotone traveling wave solutions of (1.2) with ai = 1, i ∈ I . Note that theseconclusions remain true if τ = 0, we thus confirm the conjecture about the existence ofnonmonotone traveling wave solutions of competitive systems, which was proposed by Tangand Fife [30, the last paragraph].

The rest of this paper is organized as follows. In Sect. 2, we list some preliminaries.Using contracting rectangles, the asymptotic behavior of traveling wave solutions of generalpartial functional differential equations (1.1) is established in Sect. 3, and is applied to twoexamples considered by Ma [19] and Zou [41]. In Sect. 4, we introduce the generalized upperand lower solutions and study the existence of traveling wave solutions in (1.1). In Sect. 5,we investigate the traveling wave solutions of the Lotka–Volterra system (1.2), including theexistence, nonexistence, asymptotic behavior and monotonicity. This paper ends with a briefdiscussion of our methods and results.

2 Preliminaries

In this paper, we shall use the standard partial ordering and interval notations in Rn .Namely,

if u = (u1, u2, . . . , un), v = (v1, v2, . . . , vn) ∈ Rn, then u ≥ vi f f ui ≥ vi , i ∈ I ; u >

vi f f u ≥ v but ui > vi for some i ∈ I ; u vi f f ui > vi , i ∈ I. Moreover, X will beinterpreted as follows

X = {u : u is a bounded and uniformly continuous function from R to Rn},

which is a Banach space equipped with the supremum norm ‖ · ‖. If a, b ∈ Rn with a ≤ b,

then

X[a,b] = {u ∈ X : a ≤ u(x) ≤ b, x ∈ R}.

Let u(x) = (u1(x), . . . , un(x)), v(x) = (v1(x), . . . , vn(x)) ∈ X, then u(x) ≥ v(x) impliesthat u(x) ≥ v(x) for all x ∈ R; u(x) > v(x) is interpreted as u(x) ≥ v(x) but u(x) > v(x)for some x ∈ R; and u(x) v(x) if u(x) > v(x) and for each i ∈ I, there exists xi ∈ R

such that ui (xi ) > vi (xi ).u(x) is a nonnegative, positive and strictly positive function iffu(x) ≥ 0, u(x) > 0 and u(x) 0, respectively.

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Consider the Fisher equation{∂z(x,t)∂t = d�z(x, t)+ r z(x, t) [1 − z(x, t)] ,

z(x, 0) = z(x) ∈ X[0,1](2.1)

with x ∈ R, t > 0, d > 0, r > 0 and n = 1 in the definition of X .

Lemma 2.1 For (2.1), we have the following conclusions.

(i) (2.1) admits a unique solution z(·, t) ∈ X[0,1] for all t > 0.(ii) If z(·, t), z(·, t) ∈ X for t > 0 such that that

∂z(x, t)

∂t≥ d�z(x, t)+ r z(x, t) [1 − z(x, t)] ,

∂z(x, t)

∂t≤ d�z(x, t)+ r z(x, t)

[1 − z(x, t)

],

then z(x, t)and z(x, t)are upper and lower solutions to (2.1), respectively. Furthermore,we have

z(x, t) ≥ z(x, t) ≥ z(x, t)

if z(x, 0) ≥ z(x) ≥ z(x, 0).(iii) If z(x) > 0 and c ∈ (0, 2

√dr), then

lim inft→∞ inf|x |<ct

z(x, t) = lim supt→∞

sup|x |<ct

z(x, t) = 1.

For (i) and (ii) of Lemma 2.1, we refer to Fife [9] and Ye et al. [39]. (iii) of Lemma 2.1 isthe classical theory of asymptotic spreading, see Aronson and Weinberger [3].

3 Asymptotic Behavior of Traveling Wave Solutions

Consider the following functional differential equation corresponding to (1.1)

dli (t)

dt= fi (lt ), i ∈ I, (3.1)

in which l = (l1, l2, . . . , ln) ∈ Rn, fi is defined by (1.1) and satisfies the following assump-

tions:

H1 There exists E 0 such that fi (0) = fi (E) = 0, where · denotes the constant valuedfunction in C([−τ, 0],Rn) and E = (E1, E2, . . . , En) ∈ R

n;

H2 There exist E ≥ E ≥ E ≥ 0 such that [E, E] is a positively invariant ordered intervalof (3.1), where

E = (E1, E2, . . . , En), E = (E1, E2, . . . , En);

H3 If u ∈ C([−τ, 0],Rn) and 0 ≤ u(t + s) ≤ b(0) for s ∈ [−τ, 0], then fi (ut ) :C([−τ, 0],Rn) → R is Lipschitz continuous in the sense of supremum norm, here b(0) ≤ Eis a constant vector clarified by (H4)–(H5);

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H4 There exists a one-parameter family of ordered intervals given by∑

(y) = [a(y), b(y)]such that a(0) ≤ E ≤ E ≤ b(0) and for 0 ≤ y1 ≤ y2 ≤ 1

0 ≤ a(0) ≤ a(y1) ≤ a(y2) ≤ a(1) = E = b(1) ≤ b(y2) ≤ b(y1) ≤ b(0),

where a(y) and b(y) are continuous in y ∈ [0, 1];H5

∑(y) is a strict contracting rectangle, namely, let

a(y) = (a1(y), a2(y), . . . , an(y)), b(y) = (b1(y), b2(y), . . . , bn(y)),

then for any y ∈ (0, 1) and u ∈∑(y), we have

fi (u) > 0( fi (u) < 0) if ui (0) = ai (y)(ui (0) = bi (y)), i ∈ I.

To continue our discussion, we now introduce the following definition of traveling wavesolutions of (1.1).

Definition 3.1 A traveling wave solution of (1.1) is a special solution

vi (x, t) = φi (x + ct), i ∈ I,

where c > 0 is the wave speed and � = (φ1, φ2, . . . , φn) ∈ C2(R,Rn) is the wave profile.

By the definition, � = (φ1, φ2, . . . , φn) satisfies

diφ′′i (ξ)− cφ′

i (ξ)+ f ci (�ξ ) = 0, i ∈ I, ξ ∈ R; (3.2)

in which f ci (�ξ ) : C([−cτ, 0],Rn) → R is defined by

f ci (�ξ ) = fi (�(ξ + cs)), s ∈ [−τ, 0], i ∈ I.

Using the contracting rectangles, we present the following asymptotic boundary conditionsof traveling wave solutions.

Theorem 3.2 Assume (H1)–(H5). Let � ∈ X be a positive solution of (3.2) with

E lim supξ→∞

�(ξ) ≥ lim infξ→∞ �(ξ) E . (3.3)

Then limξ→∞�(ξ) = E if �′ and �′′ are uniformly bounded.

Proof Denote

lim supξ→∞

�(ξ) = �+, lim infξ→∞ �(ξ) = �−

with

�± = (φ±1 , φ

±2 , . . . , φ

±n ).

Were the statement false, then�+ > �− holds and (3.3) implies that there exists y ∈ (0, 1)such that

a(y) ≤ �− ≤ �+ ≤ b(y).

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In particular, let y1 ∈ (0, 1) be the largest y such that the above is true, then y1 is well defined.Without loss of generality, we assume that

φ−1 = a1(y1)

with a(y) = (a1(y), a2(y), . . . , an(y)).Due to the uniform boundedness of φ′

1 and φ′′1 , the fluctuation lemma of continuous

functions implies that there exists {ξm},m ∈ N, with limm→∞ ξm = ∞ such that

lim infξ→∞ φ1(ξ) = lim

m→∞φ1(ξm) = a1(y1) ≤ lim supξ→∞

φ1(ξ) ≤ b1(y1)

and

lim infm→∞ (d1φ

′′1 (ξm)− cφ′

1(ξm)) ≥ 0.

At the same time, (H5) leads to

lim infm→∞ f c

1 (�ξm ) > 0,

and we obtain a contradiction between

lim infm→∞ (d1φ

′′1 (ξm)− cφ′

1(ξm)+ f c1 (�ξm )) > 0 (3.4)

and

lim infm→∞ (d1φ

′′1 (ξm)− cφ′

1(ξm)+ f c1 (�ξm )) = 0.

The proof is complete. ��Remark 3.3 In fact, the proof of the existence of traveling wave solutions often implies theuniform boundedness of�′ and�′′, so we will not discuss the boundedness in the followingtwo examples.

Now, we recall two scalar equations to give a simple illustration of the theorem. We firstconsider the example in Zou [41].

Example 3.4 In (3.1), let

f (ut ) = u(t − τ)[1 − u(t)].Then traveling wave solutions of the corresponding partial functional differential equationhas been established by Zou [41].

Define E = 0, E = 1 + k for any k > 0 and

a(s) = s, b(s) = (1 + k)(1 − s)+ s, s ∈ [0, 1].Then [a(s), b(s)] is a contracting rectangle of the corresponding functional differentialequation such that Theorem 3.2 is applicable to the study of traveling wave solutions. Letu(x, t) = ρ(x + ct) be a traveling wave solution of

∂u(x, t)

∂t= �u(x, t)+ ru(x, t − τ)[1 − u(x, t)], r > 0.

If ρ(ξ), ξ ∈ R, is bounded and lim infξ→∞ ρ(ξ) > 0, then limξ→∞ ρ(ξ) = 1 by Theo-rem 3.2.

If the quasimonotone condition does not hold, Ma [19] studied the traveling wave solutionsby constructing proper auxiliary systems. In particular, the traveling wave solutions of thefollowing example have been well studied.

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Example 3.5 Consider

∂w(x, t)

∂t= �w(x, t)− w(x, t)+ e2w(x, t − τ)e−w(x,t−τ) (3.5)

and denote f (w) = e2we−w.

Using the results in Smith [28, Sect. 5.2], we can give two auxiliary quasimonotone equa-tions to study the dynamics of the corresponding functional differential equations of (3.5).In addition, Ma [19] presented two auxiliary quasimonotone equations of (3.5) and obtainedthe convergence of traveling wave solutions. Let

w(x, t) = ρ(x + ct)

be a traveling wave solution of (3.5) and denote

lim infξ→∞ ρ(ξ) = ρ−, lim sup

ξ→∞ρ(ξ) = ρ+,

then the auxiliary equations in Ma [19] imply that

f (e) = e3−e ≤ ρ− ≤ ρ+ ≤ e (3.6)

and f (w) is monotone decreasing for w ∈ [e3−e, e]. If ρ+ = e, then a discussion similar tothat of (3.4) implies that f (ρ−) ≥ e by the monotonicity of f, which is impossible by (3.6).By the monotonicity of f and (3.6), we further obtain that

e3−e < f ( f (e3−e)) ≤ ρ− ≤ ρ+ ≤ f (e3−e) = f ( f (e)) < e.

Let E = f ( f (e3−e)), E = f ( f (e)), then [E, E] defines an invariant region of the corre-sponding functional differential equation of (3.5).

To continue our discussion, define

f 2(w) = f ( f (w)),

then f 2(w) satisfies the following properties:

(F1) f 2(w) is monotone increasing for w ∈ [e3−e, e];(F2) f 2(w) > w,w ∈ [e3−e, 2) while f 2(w) < w,w ∈ (2, e].

For convenience, we give the graph of f 2 in Fig. 1.Let k = 2, k1 = e3−e(< 2) and

a(s) = sk + (1 − s)k1 + εh(s), b(s) = f (sk + (1 − s)k1),

in which

2h(s) = f 2(sk + (1 − s)k1)− (sk + (1 − s)k1) > 0,

and ε < 1 is small such that

a(s) < 2, s ∈ (0, 1)

and

a(0) < f 2(e3−e) < f 2(e) = b(0).

By (F2), h(s) > 0, s ∈ (0, 1) and h(s) = 0, s = 1.

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0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

X: 2Y: 2

Fig. 1 The graph of the composition map f 2

If w(0) = a(s) with a(s) ≤ w(−τ) ≤ b(s), then

−w(0)+ f (w(−τ))≥ −a(s)+ f (b(s))

= −(sk + (1 − s)k1 + εh(s))+ f 2(sk + (1 − s)k1)

= (2 − ε)h(s) > 0, s ∈ (0, 1).

If w(0) = b(s) with a(s) ≤ w(−τ) ≤ b(s), then

−w(0)+ f (w(−τ))≤ −b(s)+ f (a(s))

< −b(s)+ f (sk + (1 − s)k1)

= 0, s ∈ (0, 1),

which implies that [a(s), b(s)] satisfies (H5). Using Theorem 3.2, limξ→∞ ρ(ξ) = 2 holds.

4 Generalized Upper and Lower Solutions

In this section, we shall study the existence of traveling wave solutions of delayed system (1.1)for any fixed c > 0, where f satisfies (H1)–(H3) in Sect. 3. We first introduce the generalizedupper and lower solutions of (3.2) as follows.

Definition 4.1 Assume that T ⊂ R contains finite points of R.Then� = (φ1, φ2, . . . , φn) ∈X[0,E] and� = (φ

1, φ

2, . . . , φ

n) ∈ X[0,E] are a pair of generalized upper and lower solutions

of (3.2) if for each ξ ∈ R \T, �′′(ξ),�

′(ξ),�′′(ξ),�′(ξ) are bounded and continuous such

that

diφ′′i (ξ)− cφ

′i (ξ)+ f c

i (�ξ ) ≤ 0 (4.1)

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with �(ξ) = (φ1(ξ), φ2(ξ), . . . , φn(ξ)) ∈ X[0,E] satisfying

φj(ξ + cs) ≤ φ j (ξ + cs) ≤ φ j (ξ + cs), φi (ξ) = φi (ξ), s ∈ [−τ, 0], i, j ∈ I,

and

diφ′′i(ξ)− cφ′

i(ξ)+ f c

i (�ξ ) ≥ 0 (4.2)

with �(ξ) = (φ1(ξ), φ2(ξ), . . . , φn(ξ)) ∈ X[0,E] satisfying

φj(ξ + cs) ≤ φ j (ξ + cs) ≤ φ j (ξ + cs), φi (ξ) = φ

i(ξ), s ∈ [−τ, 0], i, j ∈ I.

Remark 4.2 If f is quasimonotone, then Definition 4.1 is equivalent to Ma [18, Definition2.2], Wu and Zou [38, Definition 3.2]; if f is mixed quasimonotone, then Definition 4.1becomes Lin et al. [17, Definition 3.1].

For any fixed ξ ∈ R, let β > 0 be a fixed constant such that

βφi (ξ)+ f ci (�ξ )

is monotone increasing in φi (ξ), i ∈ I,� = (φ1, φ2, . . . , φn) ∈ X[0,E]. From (H3), β iswell defined.

Define constants

νi1(c) = c −√c2 + 4βdi

2di, νi2(c) = c +√c2 + 4βdi

2di, i ∈ I.

For the sake of simplicity, we denote νi1 = νi1(c), νi2 = νi2(c) without confusion. Thenνi1 < 0 < νi2 and

diν2i j − cνi j − β = 0, i ∈ I, j = 1, 2.

For � = (φ1, φ2, . . . , φn) ∈ X[0,E], define F = (F1, F2, . . . , Fn) : X → X by

Fi (�)(ξ) = 1

di (νi2 − νi1)

⎡⎢⎣

ξ∫

−∞eνi1(ξ−s) +

+∞∫

ξ

eνi2(ξ−s)

⎤⎥⎦ Li (�)(s)ds, (4.3)

herein L(�)(s) = (L1(�)(s), L2(�)(s), . . . , Ln(�)(s)) is formulated by

Li (�)(ξ) = βφi (ξ)+ f ci (�ξ ), i ∈ I.

Now, to prove the existence of (3.2), it is sufficient to seek after a fixed point of F (see Wuand Zou [38]).

Let σ < mini∈I {−νi1} be a positive constant and | · | denote the supremum norm in Rn .

Define

Bσ(R,Rn) =

{�(x) ∈ X : sup

x∈R

|�(x)| e−σ |x | < ∞}

and

|�|σ = supx∈R

|�(x)| e−σ |x |.

Then it is easy to check that Bσ (R,Rn) is a Banach space with the decay norm |·|σ .

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Before giving our main conclusion of this section, we first present some calculations. Ifφ(s) is twice differentiable, then

d

ds

[e−νi j s ((c − diνi j )φ(s)− diφ

′(s))]

= e−νi j s [−νi j (c − diνi j )φ(s)+ diνi jφ′(s)]

+e−νi j s ((c − diνi j )φ′(s)− diφ

′′(s))

= e−νi j s [−diφ′′(s)+ cφ′(s)+ βφ(s)

](4.4)

because of diν2i j − cνi j − β = 0, i ∈ I, j = 1, 2.

If a bounded function φ(s) admits continuous and bounded derivatives φ′(s) and φ′′(s)for s ∈ (a, b) with a < b, then

b∫

a

e−νi j s [−diφ′′(s)+ cφ′(s)+ βφ(s)

]ds

= e−νi j b ((c − diνi j )φ(b−)− diφ′(b−))

−e−νi j a ((c − diνi j )φ(a+)− diφ′(a+)) . (4.5)

Moreover, if ξ ∈ (a, b), then

1

di (νi2 − νi1)

⎡⎢⎣

ξ∫

a

eνi1(ξ−s) +b∫

ξ

eνi2(ξ−s)

⎤⎥⎦ (βφi (s)+ cφ′

i (s)− diφ′′i (s))ds

= 1

di (νi2 − νi1)

[eνi1(ξ−s) ((c − diνi1)φ(s)− diφ

′(s))]∣∣∣∣

ξ

a

+ 1

di (νi2 − νi1)

[eνi2(ξ−s) ((c − diνi2)φ(s)− diφ

′(s))]∣∣∣∣

b

ξ

= φi(ξ)+ e−νi2b

((c − diνi2)φ(b−)− diφ

′(b−))di (νi2 − νi1)

−e−νi1a((c − diνi1)φ(a+)− diφ

′(a+))di (νi2 − νi1)

. (4.6)

Now we state and prove the main result of this section.

Theorem 4.3 Assume that � = (φ1, φ2, . . . , φn) ∈ X[0,E] and � = (φ1, φ

2, . . . , φ

n) ∈

X[0,E] are a pair of generalized upper and lower solutions of (3.2) such that

�(ξ) ≥ �(ξ), ξ ∈ R

and

φ′i (ξ+) ≤ φ

′i (ξ−), φ′

i(ξ+) ≥ φ′

i(ξ−), ξ ∈ T, i ∈ I.

Then (3.2) has a solution � such that � ≤ � ≤ �.

Proof Let

� = {�(ξ) ∈ X : �(ξ) ≤ �(ξ) ≤ �(ξ)}.

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594 J Dyn Diff Equat (2014) 26:583–605

It is clear that � is nonempty and convex. Moreover, it is closed and bounded with respectto the decay norm | · |σ . Choose � = (φ1, φ2, . . . , φn) ∈ �, then for each fixed ξ ∈ R, thedefinition of β implies that

βφi(ξ)+ f c

i (�ξ ) ≤ βφi (ξ)+ f ci (�ξ ) ≤ βφi (ξ)+ f c

i (�ξ ) (4.7)

with �(ξ + cs) = (φ1(x + cs), φ2(x + cs), . . . , φn(x + cs)) satisfying{φ j (ξ + cs) = φ j (ξ + cs), j �= i, s ∈ [−τ, 0],φi (ξ + cs) = φi (ξ + cs), s ∈ [−τ, 0), φi (ξ) = φi (ξ)

and �(ξ + cs) = (φ1(x + cs), φ2(x + cs), . . . , φn(x + cs)) satisfying{φ j (ξ + cs) = φ j (ξ + cs), j �= i, s ∈ [−τ, 0],φi (ξ + cs) = φi (ξ + cs), s ∈ [−τ, 0), φi (ξ) = φ

i(ξ).

By (4.3)–(4.7), we obtain

Fi (�)(ξ)

≥ 1

di (νi2 − νi1)

⎡⎢⎣

ξ∫

−∞eνi1(ξ−s) +

+∞∫

ξ

eνi2(ξ−s)

⎤⎥⎦ (βφi

(s)+ cφ′i(s)− diφ

′′i(s))ds

= φi(ξ)+

∑Tj ∈T

min{eνi1(ξ−Tj ), eνi2(ξ−Tj )

}νi2 − νi1

[φ′

i(Tj+)− φ′

i(Tj−)

]

≥ φi(ξ), ξ ∈ R \ T. (4.8)

Using the continuity of Fi (�)(ξ), φi(ξ), we obtain

Fi (�)(ξ) ≥ φi(ξ), ξ ∈ R.

In a similar way, we have

Fi (�)(ξ) ≤ φi (ξ), i ∈ I, ξ ∈ R,

and

F : � → �.

Moreover, similar to those in Huang and Zou [12], Li et al. [15, Lemma 3.6] and Ma [18],F : � → � is completely continuous in the sense of the decay norm | · |σ . Using Schauder’sfixed point theorem, we complete the proof. ��Remark 4.4 Let � be a solution given by Theorem 4.3. From (4.8), we obtain

φi (ξ) > φi(ξ), ξ ∈ R, i ∈ I

if one of the following statements is true: 1) for each i ∈ I, φ′i(ξ+) > φ′

i(ξ−) for some

ξ ∈ T; 2) for each i ∈ I, (4.2) is strict on an nonempty interval. Similarly, we have

φi (ξ) < φi (ξ), ξ ∈ R, i ∈ I

if one of the following statements is true: (1) for each i ∈ I, φ′i (ξ+) < φ

′i (ξ−) for some

ξ ∈ T; (2) for each i ∈ I, (4.1) is strict on an nonempty interval.

Remark 4.5 Let� be a solution given by Theorem 4.3. Since F(�)(ξ) = �(ξ), then�′(ξ)and �′′(ξ) are uniformly bounded by the bounds of �(ξ).

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J Dyn Diff Equat (2014) 26:583–605 595

5 Traveling Wave Solutions of the Lotka–Volterra System (1.2)

In this section, we study the existence and nonexistence of traveling wave solutions of (1.2)with � = R, ai > 0, i ∈ I . We first introduce some notations.

Let u(x, t) = �(x + ct) = (ψ1(x + ct), ψ2(x + ct), . . . , ψn(x + ct)) be a traveling wavesolution of (1.2). Then �(ξ) satisfies the following functional differential system

diψ′′i (ξ)− cψ ′

i (ξ)+ riψi (ξ)

⎡⎣1 −

n∑j=1

ci j

0∫

−τψ j (ξ + cs)dηi j (s)

⎤⎦ = 0, i ∈ I, ξ ∈ R.

(5.1)

Similar to those in Li et al. [15], Lin et al. [17], Martin and Smith [21], we consider the invasionwaves of all competitors which satisfy the following asymptotic boundary conditions

limξ→−∞ψi (ξ) = 0, lim

ξ→∞ψi (ξ) = u∗i . (5.2)

Remark 5.1 From the viewpoint of population dynamics, (5.1)–(5.2) formulate the synchro-nous invasion of all competitors, we refer to Shigesada and Kawasaki [27, Chap. 7] forthe historical records of the expansion of the geographic range of several plants in NorthAmerican after the last ice age (16,000 years ago).

5.1 Existence of Traveling Wave Solutions

To construct upper and lower solutions, we define some constants. For any fixed c >

maxi∈I {2√diri }, define constants γi1 and γi2 such that 0 < γi1 < γi2 and

diγ2i1 − cγi1 + ri = diγ

2i2 − cγi2 + ri = 0 for i ∈ I. (5.3)

Assume that q > 1 holds and η satisfies

η ∈(

1, mini, j∈I

{γi2

γi1,γi1 + γ j1

γi1

}). (5.4)

Define continuous functions ψi(ξ) and ψ i (ξ) as follows

ψi(ξ) = max{eγi1ξ − qeηγi1ξ , 0}, ψ i (ξ) = min{eγi1ξ , (ai cii )

−1}, i ∈ I.

Lemma 5.2 If q > 1 is large, then ψi(ξ) and ψ i (ξ) are generalized upper and lower

solutions of (5.1).

Proof By the monotonicity, it suffices to prove that

diψ′′i (ξ)− cψ

′i (ξ)+ riψ i (ξ)

⎡⎣1 − cii aiψ i (ξ)−

n∑j=1

ci j

0∫

−τψ

j(ξ+ cs)dηi j (s)

⎤⎦≤0, i ∈ I,

(5.5)and

diψ′′i(ξ)− cψ ′

i(ξ)+riψ i

(ξ)

⎡⎣1−cii aiψ i

(ξ)−n∑

j=1

ci j

0∫

−τψ j (ξ+ cs)dηi j (s)

⎤⎦ ≥ 0, i ∈ I

(5.6)

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596 J Dyn Diff Equat (2014) 26:583–605

if ψi(ξ) and ψ i (ξ), i ∈ I, are differentiable.

If ψ i (ξ) = eγi1ξ < (ai cii )−1, then

diψ′′i (ξ)− cψ

′i (ξ)+ riψ i (ξ)

⎡⎣1 − cii aiψ i (ξ)−

n∑j=1

ci j

0∫

−τψ

j(ξ + cs)dηi j (s)

⎤⎦

≤ diψ′′i (ξ)− cψ

′i (ξ)+ riψ i (ξ)

= eγi1ξ [diγ2i1 − cγi1 + ri ]

= 0.

If ψ i (ξ) = (ai cii )−1 < eγi1ξ , then

diψ′′i (ξ)− cψ

′i (ξ)+ riψ i (ξ)

⎡⎣1 − cii aiψ i (ξ)−

n∑j=1

ci j

0∫

−τψ

j(ξ + cs)dηi j (s)

⎤⎦

≤ diψ′′i (ξ)− cψ

′i (ξ)+ riψ i (ξ)

[1 − cii aiψ i (ξ)

]

= 0,

and this completes the proof of (5.5).If ψ

i(ξ) = 0 > eγi1ξ − qeηγi1ξ , then

diψ′′i(ξ)− cψ ′

i(ξ)+ riψ i

(ξ)

⎡⎣1 − cii aiψ i

(ξ)−n∑

j=1

ci j

0∫

−τψ j (ξ + cs)dηi j (s)

⎤⎦ = 0.

If ψi(ξ) = eγi1ξ − qeηγi1ξ > 0, then

diψ′′i(ξ)− cψ ′

i(ξ)+ riψ i

(ξ)

⎡⎣1 − cii aiψ i

(ξ)−n∑

j=1

ci j

0∫

−τψ j (ξ + cs)dηi j (s)

⎤⎦

= diψ′′i(ξ)− cψ ′

i(ξ)+ riψ i

(ξ)

−ri cii aiψ2i(ξ)− riψ i

(ξ)

n∑j=1

ci j

0∫

−τψ j (ξ + cs)dηi j (s)

= −qeηγi1ξ [diη2γ 2

i1 − cηγi1 + ri ]

−ri cii aiψ2i(ξ)− riψ i

(ξ)

n∑j=1

ci j

0∫

−τψ j (ξ + cs)dηi j (s),

and the monotonicity of ψ j indicates that

−ri cii aiψ2i(ξ)− riψ i

(ξ)

n∑j=1

ci j

0∫

−τψ j (ξ + cs)dηi j (s)

≥ −ri cii aiψ2i(ξ)− riψ i

(ξ)

n∑j=1

ci jψ j (ξ)

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J Dyn Diff Equat (2014) 26:583–605 597

≥ −ri cii ai e2γi1ξ − ri

n∑j=1

ci j e(γi1+γ j1)ξ .

Therefore, we only need to verify that

− qeηγi1ξ [diη2γ 2

i1 − cηγi1 + ri ] − ri cii ai e2γi1ξ − ri

n∑j=1

ci j e(γi1+γ j1)ξ ≥ 0. (5.7)

Since q > 1, we have ξ < 0 and

eηγi1ξ > e2γi1ξ , eηγi1ξ > e(γi1+γ j1)ξ ,

which imply that (5.7) is true if

q >−ri cii ai − ri

∑nj=1 ci j

diη2γ 2i1 − cηγi1 + ri

+ 1 > 1.

Let

q = maxi∈I

{−ri cii ai − ri∑n

j=1 ci j

diη2γ 2i1 − cηγi1 + ri

}+ 2,

then (5.6) is true and we complete the proof. ��From Lemma 5.2, Theorem 4.3 and Remark 4.4, we obtain the following result.

Theorem 5.3 For each c > maxi∈I {2√diri }, (5.1) has a strictly positive solution � =

(ψ1, ψ2, . . . , ψn) such that

limξ→−∞ψi (ξ)e

−γi1ξ = 1, ψi(ξ) < ψi (ξ) < ψ i (ξ), ξ ∈ R, i ∈ I. (5.8)

5.2 Asymptotic Behavior of Traveling Wave Solutions

The following is the main conclusion of this subsection.

Theorem 5.4 Assume that (1.3) holds. If �(ξ) is formulated by Theorem 5.3, then (5.2) istrue.

Proof Note that�(x +ct) is a special classical solution of the following initial value problem⎧⎨⎩∂ui (x, t)

∂t= di�ui (x, t)+ ri ui (x, t)

[1 −∑n

j=1 ci j∫ 0−τ u j (x, t + s)dηi j (s)

],

ui (x, s) = ψi (x + cs),(5.9)

where x ∈ R, t > 0, s ∈ [−τ, 0]. The boundedness and smoothness of�(x + ct) imply thatψi (x + ct) is an upper solution to the following Fisher equation

∂ui (x, t)

∂t= di�ui (x, t)+ ri ui (x, t)

⎡⎣2 −

n∑j=1

ci j (c j j a j )−1 − ai cii ui (x, t)

⎤⎦ .

Thus Lemma 2.1 asserts that

lim infξ→∞ ψi (ξ) ≥ 2 −

n∑j=1

ci j (c j j a j )−1 > 0, i ∈ I.

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598 J Dyn Diff Equat (2014) 26:583–605

Denote

lim infξ→∞ ψi (ξ) = ψ−

i , lim supξ→∞

ψi (ξ) = ψ+i ,

then there exists s ∈ (0, 1] such that

ai (s) ≤ ψ−i ≤ ψ+

i ≤ bi (s)

with

ai (s) = su∗i , bi (s) = su∗

i + (1 − s)[(cii ai )−1 + ε], ε > 0,

a(s) = (a1(s), a2(s), . . . , an(s)), b(s) = (b1(s), b2(s), . . . , bn(s)).

By Smith [28, Lemma 7.4], there exists a constant ε > 0 such that [a(s), b(s)] defines astrictly contracting rectangle of the corresponding functional differential equations of (1.2).Applying Theorem 3.2, we complete the proof. ��

Furthermore, from the proof of Theorem 5.4, we obtain the following result.

Theorem 5.5 Assume that�(ξ) = (ψ1(ξ), ψ2(ξ), . . . , ψn(ξ)) is a strictly positive solutionto (5.1) and satisfies

0 ≤ ψi (ξ) < (ai cii )−1, i ∈ I, ξ ∈ R.

Then limξ→∞ ψi (ξ) = u∗i if (1.3) holds.

5.3 Nonexistence of Traveling Wave Solutions

Theorem 5.6 Assume that 2√

di0ri0 = maxi∈I {2√diri } for some i0 ∈ I. If c < 2

√di0ri0 ,

then (5.1) does not have a bounded positive solution � = (ψ1, ψ2, . . . , ψn) satisfying

limξ→−∞ψi (ξ) = 0, lim inf

ξ→∞ ψi0(ξ) > 0, ξ ∈ R, i ∈ I. (5.10)

Moreover, if (1.3) holds, then (5.1) does not have a strictly positive solution such that

0 ≤ ψi (ξ) < (ai cii )−1, i ∈ I, ξ ∈ R.

Proof Without loss of generality, we assume that 2√

d1r1 = maxi∈I {2√diri }. If (1.3) holds,

then (5.10) is obtained by Theorem 5.5. So we suppose that (5.10) is true. Were the statementfalse, then there exists some c′ ∈ (0, 2

√d1r1) such that (5.1) with c = c′ has a positive

solution � = (ψ1, ψ2, . . . , ψn). Then (5.10) implies that there exists M > 0 such thatψ1(ξ) = ψ1(x + c′t) satisfies

{∂w(x,t)∂t ≥ d1�w(x, t)+ r ′

1w(x, t) [1 − Mw(x, t)] ,

w(x, 0) = ψ1(x),

where 4√

d1r ′1 = 2

√d1r1 + c′. In fact, if ξ → −∞, then limξ→−∞ ψi (ξ) = 0 ensures the

admissibility of r ′1. Otherwise, lim infξ→∞ ψ1(ξ) > 0 implies the admissibility of M (may

be large but finite). By the smoothness of ψ1(x + c′t), we know that ψ1(x + c′t) is an uppersolution to the following initial value problem

{∂w(x,t)∂t = d1�w(x, t)+ r ′

1w(x, t) [1 − Mw(x, t)] ,

w(x, 0) = ψ1(x).(5.11)

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J Dyn Diff Equat (2014) 26:583–605 599

Let −2x = (2√

d1r ′1 +c′)t, then t → ∞ implies that x +c′t → −∞ such that u1(x, t) =

ψ1(x + c′t) → 0. At the same time,

−2x =(

2√

d1r ′1 + c′

)t < 4

√d1r ′

1t

and Lemma 2.1 lead to lim inf t→∞ u1(x, t) ≥ 1/M in (5.11), a contradiction occurs. Theproof is complete. ��Remark 5.7 Even if c < maxi∈I {2√

diri }, (5.1) may have a nontrivial positive solution� = (ψ1, ψ2, . . . , ψn) such that maxi∈I lim infξ→−∞ ψi (ξ) > 0. We refer to Guo andLiang [10], Huang [13] for some recent results of the corresponding undelayed eqnarrays.

Remark 5.8 Theorem 5.6 completes the discussions of Li et al. [15, Example 5.1] and Lin etal. [17, Example 5.2] by presenting the nonexistence of traveling wave solutions. Therefore, italso improves some results for undelayed systems by confirming the nonexistence of travelingwave solutions without the requirement of monotonicity, for example, Ahmad and Lazer [1]and Tang and Fife [30]. See next subsection.

5.4 Existence of Nonmonotone Traveling Wave Solutions

If τ = 0, Ahmad and Lazer [1], Tang and Fife [30] proved the existence of monotonetraveling wave solutions of (1.2). Recently, Fang and Wu [5] also confirmed the existence ofmonotone traveling wave solutions if τ is small and n = 2. In particular, Tang and Fife [30]thought that the monotonicity was a technical requirement and conjectured the existence ofnonmonotone traveling wave solutions if τ = 0 and n = 2.

In the previous section, we obtained the existence of traveling wave solutions connecting0 with u∗.Because our requirement for the auxiliary functions was very weak, we can presentsome sufficient conditions of the existence of nonmonotone traveling wave solutions if ai = 1for all i ∈ I .

By rescaling, it suffices to consider (1.2) with cii = 1, i ∈ I. Then (1.3) implies thatci j < 1, i �= j, i, j ∈ I, which further indicates that we can obtain a fixed q such thatLemma 5.2 holds for any fixed c (so η can be a constant) and for all ci j < 1, i �= j, i, j ∈ I.

Therefore, for each fixed c, there exist mi > 0 (e.g., mi = supξ∈Rψ

i(ξ)) independent of

ci j such that

supξ∈R

ψi(ξ) ≥ mi , i ∈ I.

Let u∗ be the function of ci j with i �= j, i, j ∈ I, then there exist ci j with i �= j, i, j ∈ I,such that (1.3) is true and

u∗i ≤ mi

holds for some i ∈ I. (5.8) further indicates that

supξ∈R

ψi (ξ) > u∗i ,

and the existence of nonmonotone traveling wave solutions follows from (5.2).To further illustrate our conclusions, we also give some numerical simulations. Take

{0.0001φ′′

1 (ξ)− φ′1(ξ)+ 0.1φ1(ξ) [1 − φ1(ξ)− 0.55φ2(ξ)] = 0,

0.05φ′′2 (ξ)− φ′

2(ξ)+ 0.5φ2(ξ) [1 − φ2(ξ)− 0.75φ1(ξ)] = 0

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600 J Dyn Diff Equat (2014) 26:583–605

00.2

0.40.6

0.81

0

200

400

6000

0.2

0.4

0.6

0.8

Distance x

u1(x,t)

Time t

Fig. 2 The traveling wave solution u1(x, t)

such that

k1 ≈ 0.7659, k2 ≈ 0.4255.

Then the nonmonotonic traveling wave solutions are presented in Figs. 2 and 3.

5.5 Further Results for n = 1

It is difficult to consider the existence of (5.1)–(5.2) if c = maxi∈I {2√diri }. But for n = 1,

we can obtain the existence of traveling wave solutions by passing to a limit function. Whenn = 1, (1.2) becomes

∂v(x, t)

∂t= d�v(x, t)+ rv(x, t)

⎡⎣1 −

0∫

−τv(x, t + s)dζ (s)

⎤⎦ , (5.12)

herein v ∈ R, d > 0, r > 0, x ∈ R, t > 0 and

ζ (s) is nondecreasing on [−τ, 0] and ζ (0)− ζ (−τ) = 1.

In particular, we shall suppose that τ is the real maximum delay involved in (5.12), namely,there is no τ0 < τ such that

∫ 0−τ0

dζ (s) = 1. Denote

b = ζ (0)− ζ (0−)and set

ζ(s) ={ζ (s), s ∈ [−τ, 0),

ζ (0−), s = 0.

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J Dyn Diff Equat (2014) 26:583–605 601

00.2

0.40.6

0.81

0

200

400

6000.2

0.4

0.6

0.8

1

Distance x

u2(x,t)

Time t

Fig. 3 The traveling wave solution u2(x, t)

Then (1.3) implies that

b ∈ (1/2, 1] (5.13)

and (5.13) will be imposed in this subsection.Let v(x, t) = φ(x + ct) be a traveling wave solution of (5.12). Then it satisfies

dφ′′(ξ)− cφ′(ξ)+ rφ(ξ)

⎡⎣1 − bφ(ξ)−

0∫

−τφ(ξ + cs)dζ(s)

⎤⎦ = 0, ξ ∈ R (5.14)

and the asymptotic boundary conditions

limξ→−∞φ(ξ) = 0, lim

ξ→∞φ(ξ) = 1. (5.15)

Clearly, if c > 2√

dr or c < 2√

dr , then the existence or nonexistence of (5.14)–(5.15)has been addressed by the previous results. The following result deals with the case whenc = 2

√dr .

Theorem 5.9 When c = 2√

dr , (5.12) also has a positive traveling wave solution connecting0 with 1.

Proof Let {cn} be a decreasing sequence with cn < 4√

dr and cn → 2√

dr , n → ∞. Thenfor each cn, (5.12) has a positive traveling wave solution connecting 0 with 1, denoted byφn(ξ). It follows that 0 ≤ φn(ξ) ≤ 1/b, ξ ∈ R, n ∈ N. Note that a traveling wave solutionis invariant in the sense of phase shift, so we assume that

φn(0) = 2b − 1

8b, φn(ξ) <

2b − 1

8b, ξ < 0. (5.16)

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602 J Dyn Diff Equat (2014) 26:583–605

By (5.15), we know that (5.16) is admissible. For n ∈ N, ξ ∈ R, it is evident that φn(ξ) areequicontinuous and bounded in the sense of the supremum norm. By Ascoli-Arzela lemmaand a nested subsequence argument, {φn(ξ)} has a subsequence, still denoted by {φn(ξ)},such that

φn(ξ) → φ(ξ), n → ∞for a continuous function φ(ξ). We see that the above convergence is pointwise on R and isalso uniform on any bounded interval of R.

Note that

min{eγ1(cn)(ξ−s), eγ2(cn)(ξ−s)} → min{eγ1(2√

dr)(ξ−s), eγ2(2√

dr)(ξ−s)}, n → ∞,

for any given ξ ∈ R, and the convergence in s is uniform for s ∈ R. Applying the dominatedconvergence theorem in F, we know that φ(ξ) is a fixed point of F and a solution to (5.14).

By (5.15) and (5.16), φ also satisfies

φ(0) = 2b − 1

8b, φ(ξ) ≤ 2b − 1

8b, ξ < 0, φ ∈ X[0,1/b].

Namely, φ(ξ) is a positive solution to (5.14), which is uniformly continuous for ξ ∈ R. Now,we are in a position to verify the asymptotic boundary conditions (5.15). Due to φ(0) > 0 andLemma 2.1, lim infξ→∞ φ(ξ) > 0 holds. By Theorem 5.5, we obtain limξ→∞ φ(ξ) = 1.

Define

lim supξ→−∞

φ(ξ) = φ+, lim infξ→−∞ φ(ξ) = φ−.

It is clear that

0 ≤ φ− ≤ φ+ ≤ 2b − 1

8b.

If φ− > 0, then the dominated convergence theorem in F implies that

4φ− ≥ 4φ− + φ−(1 − bφ− − (1 − b)φ+),

which is impossible by (5.16). Hence, φ− = 0 follows.If φ+ > 0, then there exist {ξ j }, j ∈ N with ξ j → −∞, j → ∞ such that φ(ξ j ) →

φ+, j → ∞. Using the uniform continuity of φ, there exists δ > 0 such that

2b − 1

8b≥ φ(ξ) ≥ φ+/2, ξ ∈ (ξ j − δ, ξ j + δ), j → ∞. (5.17)

We now return to the Fisher equation{∂z(x,t)∂t = d�z(x, t)+ r z(x, t) [1 − (1 − b)/b − bz(x, t)] ,

z(x, 0) = z(x),

of which an upper solution is φ(x + ct) if z(x) ≤ φ(x). Let z(x) satisfy

(z1) z(x) ∈ X,(z2) z(x) = φ+/2, x ∈ [−δ/2, δ/2],(z3) z(±δ) = 0 and z(x) is decreasing (increasing) if x ∈ (δ/2, δ)(x ∈ (−δ,−δ/2)),(z4) z(x) = 0 if |x | > δ.

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Because of φ(0) = 2b−18b ≥ φ+ and the uniform continuity of φ(ξ), we see that δ > 0 is

admissible. Then there exists T > 0 such that

z(0, t) >2b − 1

4b> φ+, t ≥ T (see Lemma 2.1). (5.18)

For each j ∈ N, there exists j ′ ∈ N such that ξ j ′ < ξ j − 2√

drT − 2δ. When ξ j and ξ j ′satisfy (5.17), then (5.18) and Lemma 2.1 imply that

φ(ξ j ) >2b − 1

4b> φ+,

which is impossible by (5.17) and the arbitrariness of j . Therefore, φ+ = 0. The proof iscomplete. ��

6 Discussion

Delay is a very common process in many biological and physical phenomena and manyimportant and realistic models with delay have been proposed to describe various problemsin applied subjects. The fundamental theory of delayed differential equations have beenwell-developed, we refer to the monographs of Hale and Verduyn Lunel [11] and Wu [37].It is well-known that the dynamics between the delayed and undelayed systems may besignificantly different; for instance, the delayed Logistic equation or Hutchinson equationexhibits nontrivial periodic solutions while all the nonnegative solutions of the Logisticequation converge to the positive steady state. Moreover, to study delayed systems, morecomplex phase spaces than that of the corresponding undelayed systems are required. Theinvestigation of traveling wave solutions of delayed systems is also more difficult than thatof the corresponding undelayed systems, at least the phase plane method which is powerfulin studying undelayed systems meets some difficulties in the study of delayed systems.

Of course, if a system is (local) quasimonotone, then the classical theory established formonotone semiflows is applicable and there are plentiful results. For example, the existence,nonexistence, minimal wave speed, uniqueness and stability of traveling wave solutions havebeen widely studied and many sharp results have been established, see Liang and Zhao [16],Schaaf [26], Thieme and Zhao [31], Smith and Zhao [29], and Wang et al. [35,36]. If thesystem is not quasimonotone, then the study becomes harder and some new phenomena canoccur, e.g., the existence of nonmonotone traveling wave solutions in scalar equations, seeFaria and Trofimchuk [7]. When the delay is small enough, some nice results on travelingwave solutions can also be obtained by different techniques such as exponential ordering,perturbation and so on, see Ai [2], Fang and Wu [6], Lin et al. [17], Ou and Wu [23], Wanget al. [34], and Wu and Zou [38].

However, if the delay is large, these techniques cannot deal with the traveling wave solu-tions of delayed systems including (1.2) and (5.12). In this paper, we applied generalizedupper and lower solutions to seek after the positive traveling wave solutions. Since thesesystems do not satisfy the quasimonotone condition, the limit behavior of traveling wavesolutions cannot be obtained by the monotonicity of traveling wave solutions and the domi-nated convergence theorem. In particular, for the case c = 2

√dr in (5.12), the asymptotic

behavior of traveling wave solutions cannot be considered by the techniques of monotonetraveling wave solutions in Liang and Zhao [16] and Thieme and Zhao [31]. In this paper, westudied the asymptotic behavior of traveling wave solutions of (1.2) and (5.12) by combiningthe idea of contracting rectangles with the theory of asymptotic spreading.

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Our results imply that if there exists instantaneous self-limitation effect, then the largedelays appearing in the intra-specific competition terms may not affect the persistence oftraveling wave solutions. However, very likely large delay may also lead to some significantdifferences between the traveling wave solutions of delayed and undelayed systems, e.g., thenonexistence of monotone traveling wave solutions of (5.12), which will be investigated inour future studies.

Acknowledgments The authors would like to thank an anonymous reviewer for his/her helpful commentsand Yanli Huang for her valuable suggestions. This research was partially supported by the the NationalNatural Science Foundation of China (11101194) and the National Science Foundation (DMS-1022728).

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